CN109286470B - Scrambling transmission method for active nonlinear transformation channel - Google Patents
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Abstract
The invention discloses a scrambling transmission method for an active nonlinear transformation channel. The method adopts an interference signal irrelevant to a signal to be transmitted as a known codebook, actively adds nonlinear distortion generated by the codebook into the signal to be transmitted, and transmits the codebook and a nonlinear model to a partner while transmitting the signal; the memory nonlinear characteristic is actively added before the encrypted information is sent, so that the information is hidden in a seemingly meaningless nonlinear signal, and a barrier is made to an information receiving judgment link of a non-partner, so that the judgment of a receiving code is meaningless, the information is hidden and hidden, and the interception, recovery and decoding of the information are further influenced.
Description
Technical Field
The invention relates to the technical field of communication security, in particular to a scrambling transmission method for an active nonlinear transformation channel.
Background
At the heart of the communication technology is encryption technology. The memory or memoryless nonlinear phenomenon of electronic equipment, devices and transmission links often becomes one of the key factors which restrict the improvement of system performance and influence the correct reception of signals. The encryption technology developed in the traditional cryptography mainly deals with the system with limited integers, and the chaos secret communication technology developed based on the complex dynamics has become a new research hotspot in the last decade. It has some commonalities with traditional cryptography, the most important characteristic is sensitivity to variable and parameter change, but the system used for chaotic secure communication is defined in real number. Chaotic encryption is mainly performed in the cryptology field at present, and the basic task is to find a chaotic sequence generator capable of generating better cryptology characteristics.
In fact, in addition to information security techniques, line security techniques have long been explored. It is well known that the transmission of information is to be effected by means of media. This creates a security breach and security issue since the information is transmitted through some medium. For example, in wireless communication, electromagnetic waves are emitted into the air during signal transmission, and not only can the opposite party of the communication receive the electromagnetic waves, but also other people, especially enemies can receive the electromagnetic waves. The secrecy of wired communication is relatively good, but the factors of divulgence still exist, and particularly, crosstalk and radiation of metal cables, harmonic radiation of carrier machines, eavesdropping of enemies and the like are all factors of divulgence.
In addition to information encryption (key-dependent security), one also considers line encryption. The line security technology is mainly researched on how to prevent the enemy from receiving the communication signal of the party. Line privacy may be referred to as covert privacy communications which prevents an adversary from directly acquiring signals from the present communication line (both wired and wireless) or communicating in a manner that is imperceptible to the adversary. Such as directional transmission, narrow-direction transmission, etc. in wireless communications, fiber optic communication in wired lines, and specially-made privacy cables, etc. Line privacy techniques have, however, met with limited success to date.
Most of the traditional encryption technologies do not consider the problem of judgment of unauthorized users, and the unauthorized users can receive information and judge correctly like authorized users in most cases. The security level of a secret communication system based on the cryptology theory is greatly dependent on the length of a secret key, and the fatal defect of the secret communication system is that the secret communication system clearly prompts an attacker or a listener which is important information, so that the secret communication system is easy to curiosity and notice of the attacker and increases the desire of attack cracking. Therefore, the authorized user only needs to analyze the content of the judgment symbol.
Disclosure of Invention
Aiming at the defects of the prior art, the invention aims to solve the technical problems that information leakage is easily generated in the prior secret technology of wired communication due to crosstalk and radiation of a metal cable, harmonic radiation of a carrier machine, eavesdropping of an enemy and the like, and the prior secret communication technology based on the cryptology theory clearly prompts an attacker or a listener which is important information, so that the attacker is easy to get curiosity and attention, the desire of attack cracking is increased, and the safety of the secret technology is limited.
In order to achieve the above object, the present invention provides a channel scrambling transmission method based on active memory nonlinearity, which includes the following steps:
determining a signal to be transmitted and a codebook, wherein the codebook is an interference signal irrelevant to the signal to be transmitted;
generating nonlinear distortion based on a codebook, and adding the nonlinear distortion into a signal to be sent to obtain an actively nonlinear scrambled signal, wherein the nonlinear distortion and the signal to be sent are in the same bandwidth so as to hide the signal to be sent;
and sending the actively and nonlinearly scrambled signal to a receiving end so that the receiving end demodulates the actively and nonlinearly scrambled signal based on the parameter corresponding to the nonlinear distortion to obtain the signal to be sent.
It can be understood that, a sending end sends two paths of signals to a receiving end, one path of signal is a signal containing a codebook so that the receiving end determines parameters corresponding to nonlinear distortion, and the other path of signal is a signal to be sent after nonlinear distortion scrambling, so that the receiving end demodulates the signal to be sent from the received signal after active nonlinear scrambling based on the parameters corresponding to nonlinear distortion.
Optionally, obtaining the signal after active nonlinear scrambling is realized by an active memory nonlinear transformation mathematical model;
the active memory nonlinear transformation mathematical model comprises a Volterra, Hammerstein, Wiener and memory polynomial nonlinear model;
the parameters corresponding to the nonlinear distortion are parameters of the active memory nonlinear transformation mathematical model, and the parameters comprise an order m, a memory depth n and a kernel coefficient hm(τ1,...,τn) In which τ isnThe kernel coefficient corresponding to the memory depth n is represented.
Optionally, the generating nonlinear distortion based on the codebook and adding the nonlinear distortion to the signal to be transmitted to obtain an actively nonlinearly scrambled signal specifically includes:
determining a nonlinear transfer function, wherein the nonlinear transfer function comprises a memory linear term and a memory nonlinear term, and the nonlinear transfer function is used for generating nonlinear distortion based on a codebook and adding the nonlinear distortion into a signal to be transmitted;
when the memory linear item corresponding to the signal to be transmitted is far larger than the memory nonlinear item, the signal to be transmitted is processed by the nonlinear transfer function, a part of the memory nonlinear item is subtracted on the basis of the processing result, and nonlinear distortion compensation is carried out on the result processed by the nonlinear transfer function so as to complete active nonlinear scrambling of the signal to be transmitted, wherein the nonlinear distortion corresponding to the codebook and the signal to be transmitted are in the same bandwidth after the nonlinear distortion compensation;
when the frequency of the signal to be transmitted after the nonlinear transfer function processing is overlapped with the signal bandwidth before the nonlinear transfer function processing, only the nonlinear distortion is inhibited when the nonlinear distortion compensation is carried out on the result after the nonlinear transfer function processing, and the signal to be transmitted is not influenced;
and when the memory linear term corresponding to the signal to be transmitted is the same as the signal to be transmitted, the nonlinear distortion compensation is the memory nonlinear term included in the nonlinear transfer function.
Specifically, the nonlinear transfer function of the entire receive acquisition chain can be expressed as:
y(k)＝f[x(k)]+h[x(k)]
wherein f (-) is a memory linear term, h (-) is a memory nonlinear term, x (k) represents an input signal, and the model is a nonlinear model g (-) not including a memory linear term; when f [ x (k) ] > h [ x (x) ], the digitally compensated output signal s (k) can be approximated as:
s(k)＝y(k)-g[y(k)]＝f[x(k)]+h[x(k)]-g{f[x(k)]+h[x(k)]}
≈f[x(k)]+h[x(k)]-g{f[x(k)]}
the nonlinear distortion compensation can be realized by satisfying the condition h (·) ═ g [ f (·) ];
when the frequency of the nonlinear distortion product of the whole receiving and collecting link just coincides with the frequency of the weak signal to be received, the nonlinear compensation process only inhibits the nonlinear product and does not affect the same-frequency weak signal, and when f [ x (k) ], x (k), the nonlinear distortion compensation condition can be simplified as follows: h (·) g (·).
Optionally, the demodulating, by the receiving end, the signal to be transmitted from the received signal after the active nonlinear scrambling based on the parameter corresponding to the nonlinear distortion includes:
the receiving end receives a first signal which is sent by a sending end and contains a codebook;
the receiving end receives the actively non-linearly scrambled signal sent by the sending end and marks the signal as a second signal;
the receiving end determines corresponding residual errors according to the first signal, the second signal and tap coefficients of a filter used for demodulation;
the receiving end determines a first signal matrix Z according to the first signal, determines a second signal matrix X according to the second signal, determines a tap coefficient matrix w according to the tap coefficient, determines a residual error matrix E according to the residual error,
the receiving end determines the sum of squares of the residual errors according to the first signal matrix Z, the second signal matrix X and the tap coefficient matrix w;
the receiving end differentiates the tap coefficient matrix w by the residual square sum to determine the optimal solution of the tap coefficient;
and the receiving end decodes according to the first signal, the second signal and the optimal solution of the tap coefficient to obtain a signal to be sent.
Specifically, the first signal is z (n), the second signal is x (n), and z (n) is obtained after synchronous acquisition and nonlinear module conversion of codebook-containing signals; the other path is a scrambling signal x (n) which is acquired synchronously and compensated nonlinearly;
assuming that the number of points z (N) is N, the order of the FIR filter is M, the residual error of the objective function is e (N), and the tap coefficient of the FIR filter is w, the following steps are provided:
where m is the order of the FIR filter, w (m) represents the tap coefficient of the mth order, and let:
E＝[e(n) e(n+1) e(n+2)....e(n+N-1)]T∈N×1
Z＝[z(n) z(n+1) z(n+2)....z(n+N-1)]T∈N×1
w＝[w(0) w(1) w(2)...w(M-1)]∈M×1
therefore: e-sum of residual squares of the Z-Xw objective function:
the equation is derived for w:
then w of the optimal solution is obtainedoptSatisfies the equation:
XTXwopt＝XTZ
similarly, when y (n), x (n) are both complex, the optimal solution w can be obtainedoptSatisfies the equation:
XHXwopt＝XHZ
the resulting decrypted original signal is:
wherein, wopt(m) represents an optimal solution of tap coefficients of the FIR filter.
Generally, compared with the prior art, the above technical solution conceived by the present invention has the following beneficial effects:
the invention provides a physical layer channel scrambling transmission method based on active nonlinear transformation, which actively adds nonlinear characteristics before scrambling information is sent to cover original transmission signals, thereby influencing the receiving and judgment of non-cooperative users on information, carrying out communication in a mode which is not perceived by the non-cooperative users, and being a beneficial exploration on a line encryption technology.
The scrambling transmission method of the active nonlinear transformation channel provided by the invention has a unique encryption mode, is different from the information source encryption method of the traditional cryptography, is a channel scrambling method, covers the signal to be transmitted through the nonlinear distortion of the signal irrelevant to the signal to be transmitted, has stronger confidentiality security level and is difficult to crack a secret key. Compared with the information source encryption coding method, the channel scrambling method provided by the invention has the advantages that the communication content is not easy to be intercepted and deciphered by a third party, the security level is adjustable, and different security level effects can be achieved by dynamically configuring the type, the order, the memory depth, the kernel coefficient and the like of the active memory nonlinear model.
The channel scrambling security communication system based on the active memory nonlinear transformation does not need a synchronization process similar to chaotic secure communication and is insensitive to an initial value, and the initial value is an entry point of the analog or digital chaotic secure communication which is easy to attack.
The invention provides a blind self-adaptive linearization method which can improve the dynamic range of a transmission link and also give consideration to power efficiency, and the volume, weight and power consumption of a transmission system, particularly a transmitter, are reduced. The invention has controllable complexity and expenditure.
Drawings
Fig. 1 is a flowchart of a physical layer channel scrambling transmission method based on active nonlinear transformation according to the present invention;
FIG. 2 is a schematic diagram of an active non-linear scrambling process provided by the present invention;
FIG. 3 is a schematic diagram of passive non-linearity compensation provided by the present invention;
FIG. 4 is a block diagram of an algorithm for passive non-linearity compensation provided by the present invention;
fig. 5 is a block diagram of a descrambling principle based on active non-linear scrambling provided by the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
The invention creatively provides a safety communication method based on nonlinear reverse application, for example, the information encrypted based on the traditional method is artificially added with the characteristic of actively memorizing nonlinearity before being sent, so that the information is hidden in a seemingly meaningless nonlinear signal, and the information receiving judgment link of an unauthorized user is obstructed, so that the judgment of a received code is meaningless, and the recovery and the interception of the information are further influenced.
The invention provides a new channel scrambling transmission technology, which creatively provides a safety communication mechanism based on nonlinear reverse application, and information encrypted based on a traditional method is artificially added with an active memory nonlinear characteristic before being sent, so that the information is hidden in a seemingly meaningless nonlinear signal, a non-cooperative party cannot identify whether the information is an encryption means or is caused by the nonlinear characteristic of a transmission link in a short time, and the data transmission safety of a communication system is improved.
Specifically, the invention provides an active nonlinear transformation channel scrambling transmission technology, which comprises the following steps:
(1) the original signal is mapped to a constellation diagram after being modulated, and the modulated signal is sent to an encryption channel after being processed by active memory nonlinear transformation, conditioning and the like;
(2) determining a transfer function of active memory nonlinear transformation, wherein the transfer function comprises a function expression of an amplitude transfer function, namely an amplitude-amplitude (AM-AM) curve, and a function expression of a phase transfer function, namely an amplitude-phase (AM-PM) curve;
(3) receiving a nonlinear scrambling signal output by sending equipment, and conditioning the scrambling signal and then carrying out subsequent processing;
(4) adopting an interference signal irrelevant to a signal to be sent as a known codebook, and actively adding nonlinear distortion generated by the codebook into the signal to be sent;
(5) regarding the receiving and collecting system as a memory nonlinear transmission system, wherein the transfer function of the nonlinear system is f (); inverse function f of the non-linear function-1(.) is cascaded to form a two-stage cascaded nonlinear system, and the output of the two-stage cascaded nonlinear system is the original input signal;
(6) the method comprises the steps of obtaining z (n) by carrying out synchronous acquisition and nonlinear module conversion on a known codebook-containing signal, obtaining a signal x (n) by carrying out synchronous acquisition and nonlinear compensation on a scrambling-containing signal, and carrying out nonlinear decryption on the signal x (n) to obtain a decrypted original signal;
(7) and finally, demodulating and recovering the original signal by taking the parameters of the active memory nonlinear model as a decryption key.
The following is a detailed description of specific embodiments.
Fig. 1 is a scrambling transmission method for an active nonlinear transformation channel provided by the present invention, as shown in fig. 1, the steps include:
(1) signal modulation and transmission
The original signal is mapped to a constellation diagram after modulation. The modulation signal is sent to an encryption channel after being processed by active memory nonlinear transformation, conditioning and the like. The invention can be applied to the encrypted transmission of digital signals and analog signals in wireless channels and wired channels.
(2) Active memory nonlinear transformation
Mathematical models that can be used for active memory nonlinear transformation include, but are not limited to, nonlinear models such as Volterra, Hammerstein, Wiener, memory polynomial, etc., and parameters of the nonlinear models are determined: including order m, memory depth n and kernel coefficient hm(τ1,...,τn) In which τ isnThe kernel coefficient corresponding to the memory depth n is represented. Thereby determining the transfer function of the active memory nonlinear transformation, including the amplitude transfer function, i.e. amplitude-amplitudeA functional expression of a degree (AM-AM) curve and a functional expression of a phase transfer function, i.e. an amplitude-phase (AM-PM) curve.
(3) Signal reception conditioning
And receiving the nonlinear scrambling signal output by the sending equipment, and conditioning the scrambling signal and then carrying out subsequent processing.
(4) Signal channel scrambling
As shown in fig. 2, the idea of the active nonlinear transformation channel scrambling transmission technology is to use an interference signal unrelated to a signal to be transmitted as a known codebook, and actively add nonlinear distortion generated by the codebook to the signal to be transmitted.
(5) Passive non-linear compensation
As shown in fig. 3, the basic principle of the non-linear blind identification compensation is; regarding the receiving and collecting system as a memory nonlinear transmission system, wherein the transfer function of the nonlinear system is f (); inverse function f of the non-linear function-1(.) is cascaded thereafter to form a two-stage cascaded nonlinear system, the output of which is the original input signal.
The generating of the nonlinear distortion based on the codebook and adding the nonlinear distortion into the signal to be transmitted to obtain the signal after active nonlinear scrambling specifically includes:
determining a nonlinear transfer function, wherein the nonlinear transfer function comprises a memory linear term and a memory nonlinear term, and the nonlinear transfer function is used for generating nonlinear distortion based on a codebook and adding the nonlinear distortion into a signal to be transmitted; when the memory linear item corresponding to the signal to be transmitted is far larger than the memory nonlinear item, the signal to be transmitted is processed by the nonlinear transfer function, a part of the memory nonlinear item is subtracted on the basis of the processing result, and nonlinear distortion compensation is carried out on the result processed by the nonlinear transfer function so as to complete active nonlinear scrambling of the signal to be transmitted, wherein the nonlinear distortion corresponding to the codebook and the signal to be transmitted are in the same bandwidth after the nonlinear distortion compensation; when the frequency of the signal to be transmitted after the nonlinear transfer function processing is overlapped with the signal bandwidth before the nonlinear transfer function processing, only the nonlinear distortion is inhibited when the nonlinear distortion compensation is carried out on the result after the nonlinear transfer function processing, and the signal to be transmitted is not influenced; and when the memory linear term corresponding to the signal to be transmitted is the same as the signal to be transmitted, the nonlinear distortion compensation is the memory nonlinear term included in the nonlinear transfer function.
As shown in fig. 4, the nonlinear transfer function of the whole receiving acquisition chain can be expressed as:
y(k)＝f[x(k)]+h[x(k)]
wherein f (-) is a memory linear term, and h (-) is a memory nonlinear term. The nonlinear compensation model is a nonlinear model g (-) that does not contain a memory linear term. When f [ x (k) ] > h [ x (x) ], the digitally compensated output signal can be approximated as:
s(k)＝y(k)-g[y(k)]＝f[x(k)]+h[x(k)]-g{f[x(k)]+h[x(k)]}
≈f[x(k)]+h[x(k)]-g{f[x(k)]}
it can be seen that the nonlinear distortion compensation can be achieved as long as the condition h (·) ═ g [ f (·) is satisfied.
The above equation shows that when the frequency of the nonlinear distortion product of the whole receiving and collecting link just coincides with the frequency of the weak signal to be received, the nonlinear compensation process only inhibits the nonlinear product and does not affect the same-frequency weak signal.
When f [ x (k) ] ═ x (k), the above-described condition for nonlinear distortion compensation can be simplified as follows: h (·) g (·).
(6) Signal non-linear decryption
The receiving end demodulates the signal to be transmitted from the received signal after the active nonlinear scrambling based on the parameter corresponding to the nonlinear distortion to obtain the signal to be transmitted, and the method comprises the following steps:
the receiving end receives a first signal which is sent by a sending end and contains a codebook; the receiving end receives the actively non-linearly scrambled signal sent by the sending end and marks the signal as a second signal; the receiving end determines corresponding residual errors according to the first signal, the second signal and tap coefficients of a filter used for demodulation; the receiving terminal determines a first signal matrix Z according to the first signal, determines a second signal matrix X according to the second signal, determines a tap coefficient matrix w according to the tap coefficient, determines a residual error matrix E according to the residual error, and determines a residual error square sum according to the first signal matrix Z, the second signal matrix X and the tap coefficient matrix w; the receiving end differentiates the tap coefficient matrix w by the residual square sum to determine the optimal solution of the tap coefficient; and the receiving end decodes according to the first signal, the second signal and the optimal solution of the tap coefficient to obtain a signal to be sent.
Specifically, the first signal is z (n), the second signal is x (n), and as shown in fig. 5, a signal containing a known codebook is set and is subjected to synchronous acquisition and nonlinear module conversion to obtain z (n); the other path is a scrambling signal x (n) which is acquired synchronously and compensated nonlinearly.
Let the number of points z (N) be N and the order of the FIR filter be M. Let the residual of the objective function be e (n) and the tap coefficients of the FIR filter be w.
Then there are:
order to
E＝[e(n) e(n+1) e(n+2)....e(n+N-1)]T∈N×1
Z＝[z(n) z(n+1) z(n+2)....z(n+N-1)]T∈N×1
w＝[w(0) w(1) w(2)...w(M-1)]∈M×1
Therefore: sum of squares of residuals of the Z-Xw objective function
The equation is derived for w:
then w of the optimal solution is obtainedoptSatisfies the equation:
XTXwopt＝XTZ
similarly, when y (n), x (n) are both complex, the optimal solution w can be obtainedoptSatisfies the equation:
XHXwopt＝XHZ
the resulting decrypted original signal is:
(7) decrypting, demodulating and recovering original signal
And finally, demodulating and recovering the original signal by taking the parameters of the active memory nonlinear model as a decryption key.
The invention provides a channel scrambling transmission method based on active nonlinear change, which is different from the information source encryption method of the traditional cryptography and is a method for channel scrambling; the non-cooperative party can not judge, but can not decipher, so that the safety is high; the method is realized in a digital domain, does not need to simulate the strict circuit symmetry requirement in chaotic masking, and is less sensitive to thermal noise and the like.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (2)
1. An active nonlinear transformation channel scrambling transmission method is characterized by comprising the following steps:
determining a signal to be transmitted and a codebook, wherein the codebook is an interference signal irrelevant to the signal to be transmitted;
generating nonlinear distortion based on a codebook, and adding the nonlinear distortion into a signal to be sent to obtain an actively nonlinear scrambled signal, wherein the nonlinear distortion and the signal to be sent are in the same bandwidth so as to hide the signal to be sent;
sending the actively non-linearly scrambled signal to a receiving end so that the receiving end demodulates the actively non-linearly scrambled signal based on the parameter corresponding to the non-linear distortion to obtain the signal to be sent;
the generating nonlinear distortion based on the codebook and adding the nonlinear distortion into a signal to be transmitted to obtain an actively nonlinearly scrambled signal specifically includes: determining a nonlinear transfer function, wherein the nonlinear transfer function comprises a memory linear term and a memory nonlinear term, and the nonlinear transfer function is used for generating nonlinear distortion based on a codebook and adding the nonlinear distortion into a signal to be transmitted; when the memory linear item corresponding to the signal to be transmitted is far larger than the memory nonlinear item, the signal to be transmitted is processed by the nonlinear transfer function, a part of the memory nonlinear item is subtracted on the basis of the processing result, and nonlinear distortion compensation is carried out on the result processed by the nonlinear transfer function so as to complete active nonlinear scrambling of the signal to be transmitted, wherein the nonlinear distortion corresponding to the codebook and the signal to be transmitted are in the same bandwidth after the nonlinear distortion compensation; when the frequency of the signal to be transmitted after the nonlinear transfer function processing is overlapped with the signal bandwidth before the nonlinear transfer function processing, only the nonlinear distortion is inhibited when the nonlinear distortion compensation is carried out on the result after the nonlinear transfer function processing, and the signal to be transmitted is not influenced; when the memory linear term corresponding to the signal to be transmitted is the same as the signal to be transmitted, the nonlinear distortion compensation is the memory nonlinear term included in the nonlinear transfer function;
the receiving end demodulates the signal to be transmitted from the received signal after the active nonlinear scrambling based on the parameter corresponding to the nonlinear distortion to obtain the signal to be transmitted, and the demodulation comprises the following steps: the receiving end receives a first signal which is sent by a sending end and contains a codebook; the receiving end receives the actively non-linearly scrambled signal sent by the sending end and marks the signal as a second signal; the receiving end determines corresponding residual errors according to the first signal, the second signal and tap coefficients of a filter used for demodulation; the receiving terminal determines a first signal matrix Z according to the first signal, determines a second signal matrix X according to the second signal, determines a tap coefficient matrix w according to the tap coefficient, determines a residual error matrix E according to the residual error, and determines a residual error square sum according to the first signal matrix Z, the second signal matrix X and the tap coefficient matrix w; the receiving end differentiates the tap coefficient matrix w by the residual square sum to determine the optimal solution of the tap coefficient; and the receiving end decodes according to the first signal, the second signal and the optimal solution of the tap coefficient to obtain a signal to be sent.
2. The active nonlinear conversion channel scrambling transmission method of claim 1, wherein obtaining the actively nonlinear scrambled signal is achieved by an active memory nonlinear conversion mathematical model;
the active memory nonlinear transformation mathematical model comprises a Volterra, Hammerstein, Wiener and memory polynomial nonlinear model;
the parameters corresponding to the nonlinear distortion are parameters of the active memory nonlinear transformation mathematical model, and the parameters comprise an order m, a memory depth n and a kernel coefficient hm(τ1,...,τn) In which τ isnThe kernel coefficient corresponding to the memory depth n is represented.
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